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Estimating Response Modeling Methodology models

Focus Article

Haim Shore

Published Online: Feb 10 2012

DOI: 10.1002/wics.1199

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Abstract An advanced review of Response Modeling Methodology (RMM) has recently summarized RMM core philosophy, modeling approach, and allied statistical expressions. This focus article complements the earlier review by presenting a step‐by‐step guide to estimating RMM models. The estimation procedure comprises two stages: first the median is estimated and then the rest of the RMM parameters are estimated. Three estimation procedures are presented for the latter stage: maximum likelihood, two‐moment matching, and nonlinear quantile regression. The three estimation methods, as applied to RMM, are first expounded and then demonstrated via a numerical example, using Monte‐Carlo simulated data from a γ distribution and an L4 orthogonal array design. Comparisons with generalized linear modeling and estimation, assuming γ distribution (correctly) and inverse Gaussian distribution (incorrectly), are given. A brief introduction to RMM is also provided. WIREs Comput Stat 2012, 4:323–333. doi: 10.1002/wics.1199 This article is categorized under: Statistical and Graphical Methods of Data Analysis > Density Estimation Statistical Learning and Exploratory Methods of the Data Sciences > Modeling Methods Statistical Models > Simulation Models

References

Shore, H. Response modelling methodology (RMM). An advanced review. WIREs Comput Stat 2011, 3:357–372.
Shore, H. Response Modeling Methodology (RMM): Empirical Modeling for Engineering and Science. Singapore: World Scientific Publishing Co. Ltd.; 2005, 435.
Koenker, R. Package “quantreg”—quantile regression and related method. V. 4.71. Available at: http://cran.r‐project.org/web/packages/quantreg/quantreg.pdf. (Accessed September 2011).
SAS. The QUANTREG Procedure (Experimental). Document associated with this procedure may be found at: http://support.sas.com/rnd/app/papers/quantreg.pdf.
Shore, H. Comparison of generalized λ distribution (GLD) and Response Modeling Methodology (RMM) as general platforms for distribution fitting. Commun Stat Theory Methods 2007, 36:2805–2819.
Haldane, JBS. Mode and median of a nearly normal distribution with given cumulants. Biometrika 1942, 32:294.
Stuart, A, Ord, JK. Kendall`s Advanced Theory of Statistics. V. 1: Distribution Theory. 5th ed. London: Charles Griffin %26 Co.; 1987, 604.
Shore, H. Simple general approximations for a random variable and its inverse distribution function based on linear transformations of a non skewed variate. SIAM J Sci Stat Comput 1986, 7:1–23.
Shore, H. General control charts for attributes. IIE Trans Qual Reliab Eng 2000, 32:1149–1160.
Koenker, R. Quantile Regression (Econometric Society Monographs). Kindle ed. New York, NY: Cambridge University Press, 2005.

Author Notes

In this note we wish to deliver some practical advice about modeling with RMM, as expounded in the recent two papers published in this journal:

1. In modeling with RMM the parameter b appears as (1/b). Searching for an estimate of b, the computer may issue a warning about division by zero. To avoid the problem, it is good practice to replace b by (b+0.00000001), say. This will solve the problem without introducing a meaningful bias in the estimate of b.

2. In eq. 6, assuming that y represents the original data (before adding a location parameter), log(y) should be replaced by log(y-L₀).

3. For modeling with a single regressor variable, two very effective RMM models are (y is the response and x is the regressor variable):

where {a,b,c,d,M_X,M_Y,L₀} are parameters that need estimation. The parameters {M_X,M_Y} may be estimated from the data medians before estimating the other parameters. Alternatively, one may force the fitted model to pass through the data point {M_x,M_Y}. This data point may be selected arbitrarily from the sample (provided it is non-deviant), or it may be selected as the data point that delivers best value for the objective function (minimized during estimation). The location parameter, L₀, should be used only if response data are far from zero or if they include both positive and negative values. Often L₀ may be estimated by the smallest response data.

4. Objective function used to estimate the above models may be:

- MAD (mean absolute deviation), if the model is used to estimate the median;

- MSE (mean squared error), if the above model is used to estimate the mean.

Originally, the above RMM models were intended to estimate the median.

5. From our cumulative experience estimates of the parameters {a,c} may be highly linearly correlated. Therefore it is good practice, while estimating the above models, to examine correlations between the estimated parameters. This may help reduce the number of parameters that need estimation (by expressing one of the correlated estimates in terms of the other).

6. Adding constraints on the intervals where optimal parameters' estimates are searched is good practice. For example, if obtained estimates are very large numbers (in their absolute values) one should attempt imposing constraints that might deliver better solutions. From our experience, estimates of parameters {a,b,c,d} should be not far from zero.

Haim Shore

May, 7th, 2012

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